Entanglement dynamics in quantum many-body systems

TL;DR

Problem: understand and quantify entanglement dynamics in quantum many-body systems and develop a practical measurement scheme. Approach: relate entanglement growth to the real-space spreading of locally supported basis operators under local Hamiltonians, derive a toy model predicting linear growth, and reinterpret the Rényi entropy via a Loschmidt-echo in a replicated system. Key results: for a subregion of volume $N_A=\gamma_d r_A^d$, the second Rényi entropy grows as $S_2(t)=\gamma_d((r_A+v t)^d-r_A^d)\log 2$ (with an initial linear regime) and saturates at long times; an experimental protocol using a quantum switch and replica trick enables access to $S_n(t)$ from a local measurement. Significance: provides a concrete, local-measurement pathway to probe entanglement dynamics and ergodicity vs localization, with direct relevance to detecting MBL and testing operator-spreading theories; the approach connects microscopic operator dynamics to macroscopic entanglement growth and offers a practical route for experimental verification.

Abstract

The dynamics of entanglement has recently been realized as a useful probe in studying ergodicity and its breakdown in quantum many-body systems. In this paper, we study theoretically the growth of entanglement in quantum many-body systems and propose a method to measure it experimentally. We show that entanglement growth is related to the spreading of local operators in real space. We present a simple toy model for ergodic systems in which linear spreading of operators results in a universal, linear in time growth of entanglement for initial product states, in contrast with the logarithmic growth of entanglement in many-body localized (MBL) systems. Furthermore, we show that entanglement growth is directly related to the decay of the Loschmidt echo in a composite system comprised of several copies of the original system, in which connections are controlled by a quantum switch (two-level system). By measuring only the switch's dynamics, the growth of the Rényi entropies can be extracted. Our work provides a way of understanding entanglement dynamics in many-body systems, and to directly measure its growth in time via a single local measurement.

Figures (2)

• Figure 1: Set-up to measure the growth of the $n$-th Rényi entropy $S_n(t)$. Here $n = 3$ and we have shown it for a $d = 1$ chain. There is a quantum switch in the middle of the set-up which governs tunneling between different subsystems. (a) When the quantum switch is in the state $| \uparrow\rangle$, tunneling between $A_i$ and $B_i$ is allowed while tunneling between $A_i$ and $B_{i+1}$ is prohibited. (b) When the quantum switch is in the state $| \downarrow\rangle$, the allowed and prohibited tunnelings are swapped. A composite state in an $n$-copy product state of the chains and in a superposition of $| \uparrow\rangle$ and $| \downarrow\rangle$ of the quantum switch will have two parts evolving in time differently according to the quantum switch; measuring $\sigma^x(t)$ of the quantum switch gives the Loschmidt echo and hence the $n$-th Rényi entropy.
• Figure 2: (Color online) (a) The support of a typical basis operator $\mathcal{O}_A$. Here $\Lambda$ is a $d=2$ square lattice, and the subregion $A$ is a circle of radius $r_A$ demarcated by the blue circle. Black sites represent the presence of a non-trivial operator, $\sigma_i^x, \sigma_i^y$ or $\sigma_i^z$, while white sites represent the presence of an identity operator $\mathbb{I}_i$. One can see clusters of white sites within the circle. A site within a cluster is $\sim O(1)$ sites away from a black site. (b) Under time evolution, each black site spreads with velocity $v$, and so the clusters quickly get filled up (red sites) in time $vt \sim O(1)$, and simultaneously the operator also grows in physical size to become a ball of radius $r_A+vt$.